### Table 2: Syntax and semantics of the DLP description logic

1999

"... In PAGE 3: ... The more expressive logic implemented by DLP includes proposi- tional dynamic logic (PDL) [16], augmenting PDL with number restrictions on atomic roles. The syntax and semantics of the more expressive logic is given in Table2 . In this table A is 2We augmented FaCT and DLP with an interface that allows these systems to perform directly as reasoners for various propositional modal logics.... ..."

Cited by 48

### Table 2: Syntax and semantics of the DLP description logic

"... In PAGE 3: ... The more expressive logic implemented by DLP includes proposi- tional dynamic logic (PDL) [16], augmenting PDL with number restrictions on atomic roles. The syntax and semantics of the more expressive logic is given in Table2 . In this table A is 2We augmented FaCT and DLP with an interface that allows these systems to perform directly as reasoners for various propositional modal logics.... ..."

### Table 2. Parts of Description Logic reasoning rules

### Table 1: Constructors in First-Order Description Logics

1999

"... In PAGE 3: ... The for- mer are interpreted as subsets of a given domain, and the latter as binary relations on the domain. Table1 lists constructors that allow one to build #28complex#29 concepts and roles from #28atomic#29 concept names and role names. For instance, the concept Man u9Child:#3Eu8Child:Human denotes the set of... In PAGE 3: ...Table 1: Constructors in First-Order Description Logics Description logics di#0Ber in the constructions they admit. By combining constructors taken from Table1 , two well-known hierarchies of description logics may be obtained. The logics we consider here are extensions of FL , ; this is the logic with #3E, ?, universal quanti#0Ccation, conjunction and un- quali#0Ced existential quanti#0Ccation 9R:#3E.... In PAGE 4: ... For instance, FLEU , is FL , with #28full#29 existential quanti#0Ccation and disjunction. Description logics are interpreted on interpretations I =#28#01 I ; #01 I #29, where #01 I is a non-empty domain, and #01 I is an interpretation function assigning subsets of #01 I to concept names and binary relations over #01 I to role names; complex concepts and roles are interpreted using the recipes speci#0Ced in Table1 . The semantic value of an expression E in an interpretation I is simply the set E I .... In PAGE 4: ...ome page at http:#2F#2Fdl.kr.org#2Fdl#2F. 3 De#0Cning Expressive Power In this section we de#0Cne our notion of expressive power, and explain our method for determining the expressivepower of a given description logic. Our aim in this paper is to determine the expressive power of concept expressions of every extension of FL , and AL that can be de#0Cned using the constructors in Table1 . Wesay that a logic L 1 is at least as expressive as a logic L 2 if for every concept expression in L 2 there is an equivalent concept expression in L 1 ; notation: L 2 #14 L 1 .... In PAGE 4: ... First, item 1 is next to trivial. The semantics given in Table1 induces translations #28#01#29 #1C and #28#01#29 #1B taking concepts and roles, respectively, to formulas in a #0Crst-order language whose signature consists of unary predicate symbols corresponding... In PAGE 7: ... Hence, ALC #3C ALCR, ALCN, ALCRN. a Now, what do we need to do to adapt the above result for other exten- sions of FL , de#0Cned by Table1 ? For logics less expressive than ALC we... In PAGE 8: ... We #0Crst consider the `minimal apos; logic FL , ,char- acterize its concepts semantically, and use the characterization to separate FL , from richer logics. After that, we treat each of the constructors in Table1 that are not in FL , , and examine which changes are needed to characterize the concepts de#0Cnable in the resulting logics. This is followed by a brief section in which we consider combinations of constructors.... In PAGE 18: ... FL , FLE , FLU , AL FLN , FLR , FLEU , ALE FLEN , FLER , ALU FLUN , FLUR , ALN ALR FLNR , ALC FLEUN , FLEUR , ALEN ALER FLENR , ALUN ALUR FLUNR , ALNR ALCN ALCR FLEUNR , ALENR ALUNR ALCNR Figure 2: Classifying Description Logics Several comments are in order. First, the diagram does not mention all possible combinations of the constructors listed in Table1 . The reason for... In PAGE 21: ... A second important di#0Berence between Baader apos;s work and ours lies in the type of results that have been obtained. Baader only establishes a small number of separation results, whereas we provide a complete classi#0Ccation of all languages de#0Cnable using the constructors in Table1 . More importantly, our separation results are based on semantic characterizations; this gives a deeper insightinto the properties of logics than mere separation results.... In PAGE 35: ... B.6 Classifying an Arbitrary Description Logic To obtain a characterization of an arbitrary description logic #28de#0Cned from Table1 #29, simply combine the observations listed in Sections B.... ..."

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### Table 1. (Taken from [245]) Semantics of Horn and Disjunctive Logic Programs

"... In PAGE 10: ... There have been a large number of papers devoted to de ning the semantics of these databases. These semantics, drawn from [245], are summarized in the second and third columns of Table1 . The most prominent of these for the Horn case, are the well-founded semantics of Van Gelder, Ross and Schlipf [137], and the stable semantics of Gelfond and Lifschitz [139].... In PAGE 17: ... The D-WFS semantics of Brass and Dix [45] is also of considerable interest as it permits a general approach to bottom-up computation in disjunctive programs. The last two columns of Table1 summarize work accomplished in the seman- tics of DDDBs, and is drawn from [245]. See, also, the paper by Eiter, Leone and Sacc a [108], in this collection, that summarizes complexity results for par- tial models for disjunctive databases.... ..."

### Table 2. Syntax and semantics of the DLP description logic

1999

"... In PAGE 3: ... The more expressive logic implemented by DLP includes proposi- tional dynamic logic (PDL) [16], augmenting PDL with number restrictions on atomic roles. The syntax and semantics of the more expressive logic is given in Table2 . In this table A is an atomic concept, a29 and a31 are concept expressions, P is an atomic role, and a20 and a27 are arbitrary roles.... ..."

Cited by 48

### Table 3. Reasoning about location using Description Logic

### Table 1: Constructors in First-Order Description Logics

"... In PAGE 2: ... The for- mer are interpreted as subsets of a given domain, and the latter as binary relations on the domain. Table1 lists constructors that allow one to build (complex) concepts and roles from (atomic) concept names and role names.... In PAGE 3: ...Table 1: Constructors in First-Order Description Logics Description logics di er in the constructions they admit. By combining constructors taken from Table1 , two well-known hierarchies of description logics may be obtained. The logics we consider here are extensions of FL?; this is the logic with gt;, ?, universal quanti cation, conjunction and un- quali ed existential quanti cation 9R: gt;.... In PAGE 3: ... For instance, FLEU? is FL? with (full) existential quanti cation and disjunction. Description logics are interpreted on interpretations I = ( I; I), where I is a non-empty domain, and I is an interpretation function assigning subsets of I to concept names and binary relations over I to role names; complex concepts and roles are interpreted using the recipes speci ed in Table1 . The semantic value of an expression E in an interpretation I is simply the set EI.... In PAGE 4: ... First, item 1 is next to trivial. The semantics given in Table1 induces translations ( ) and ( ) taking concepts and roles, respectively, to formulas in a rst-order language whose signature consists of unary predicate symbols corresponding to atomic concepts names, and binary predicate symbols corresponding to... In PAGE 7: ... Hence, ALC lt; ALCR, ALCN, ALCRN. a Now, what do we need to do to adapt the above result for other exten- sions of FL? de ned by Table1 ? For logics less expressive than ALC we can not just use bisimulations, as such logics lack negation or disjunction, and these are automatically preserved under bisimulations; moreover, the proof of Theorem 3.3 uses the presence of the booleans in an essential way.... In PAGE 8: ...Table1 that are not in FL?, and examine which changes are needed to characterize the resulting logics. This is followed by a section in which we consider combina- tions of constructors.... In PAGE 20: ...7.6 Classifying an Arbitrary Description Logic To obtain a characterization of an arbitrary description logic (de ned from Table1 ), somply combine the observations listed in Sections 4.... In PAGE 20: ... Several comments are in order. First, the diagram does not mention all possible combinations of the constructors listed in Table1 . The reason for... ..."

### Table 1: This table indicates the number of disjuncts and the coverage of disjuncts in de nitions induced by

"... In PAGE 1: ... A disjunct is called \small quot; if its coverage is low. Table1 (column 3) shows the coverage of disjuncts in induced de nitions. There are several reasons for paying special attention to the methods by which small dis- juncts are created.... In PAGE 7: ... Cov- erage, in this case, may be de ned in several ways. Table1 reports the total number of examples matched by a disjunct. This gives larger values for coverage than alternative de nitions, such as the number of examples matched by the disjunct and no other disjunct.... In PAGE 7: ... Unlike all the other systems surveyed, which are nonincre- mental and rule-based, Protos does not attempt to minimize the number of disjuncts in the de nitions it produces. Consequently, Protos apos;s de nitions often have many disjuncts with a coverage of 1 (there were an average of 10 such disjuncts in each de nition described in Table1 ). To prevent the Protos de nitions from skewing the data, these disjuncts have been ignored.... ..."

### Table 8. Extension of pi for number restrictions and the identity role operator.

2004

"... In PAGE 18: ... Decidable description logics and first-order fragments without identity 3.6 Number restrictions, identity roles and equality So far we have excluded those operators of DL whose translation into first-order logic requires the presence of equality, namely, number restrictions and identity roles; see Table8 . As pointed out before, in the presence of number restrictions and identity roles, the translation of a knowledge base needs to incorporate in- equations representing the unique name assumption.... ..."

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